3.1.89 \(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx\) [89]

Optimal. Leaf size=210 \[ \frac {256 a^2 (13 A-3 B) c^6 \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \]

[Out]

256/15015*a^2*(13*A-3*B)*c^6*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(5/2)+64/3003*a^2*(13*A-3*B)*c^5*cos(f*x+e)^5/f/(
c-c*sin(f*x+e))^(3/2)-2/13*a^2*B*c^2*cos(f*x+e)^5*(c-c*sin(f*x+e))^(3/2)/f+8/429*a^2*(13*A-3*B)*c^4*cos(f*x+e)
^5/f/(c-c*sin(f*x+e))^(1/2)+2/143*a^2*(13*A-3*B)*c^3*cos(f*x+e)^5*(c-c*sin(f*x+e))^(1/2)/f

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2935, 2753, 2752} \begin {gather*} \frac {256 a^2 c^6 (13 A-3 B) \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 c^5 (13 A-3 B) \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 (13 A-3 B) \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 (13 A-3 B) \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

(256*a^2*(13*A - 3*B)*c^6*Cos[e + f*x]^5)/(15015*f*(c - c*Sin[e + f*x])^(5/2)) + (64*a^2*(13*A - 3*B)*c^5*Cos[
e + f*x]^5)/(3003*f*(c - c*Sin[e + f*x])^(3/2)) + (8*a^2*(13*A - 3*B)*c^4*Cos[e + f*x]^5)/(429*f*Sqrt[c - c*Si
n[e + f*x]]) + (2*a^2*(13*A - 3*B)*c^3*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(143*f) - (2*a^2*B*c^2*Cos[e +
f*x]^5*(c - c*Sin[e + f*x])^(3/2))/(13*f)

Rule 2752

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rule 2753

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-b)*(
g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Dist[a*((2*m + p - 1)/(m + p)), Int
[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2,
0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

Rule 2935

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x
] + Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; F
reeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p +
1, 0]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac {1}{13} \left (a^2 (13 A-3 B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac {1}{143} \left (12 a^2 (13 A-3 B) c^3\right ) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac {1}{429} \left (32 a^2 (13 A-3 B) c^4\right ) \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac {\left (128 a^2 (13 A-3 B) c^5\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx}{3003}\\ &=\frac {256 a^2 (13 A-3 B) c^6 \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1355\) vs. \(2(210)=420\).
time = 6.45, size = 1355, normalized size = 6.45 \begin {gather*} \frac {(7 A-2 B) \cos \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(4 A+B) \cos \left (\frac {3}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{32 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(22 A-7 B) \cos \left (\frac {5}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{160 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(A-4 B) \cos \left (\frac {7}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {A \cos \left (\frac {9}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{48 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(2 A-3 B) \cos \left (\frac {11}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{352 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {B \cos \left (\frac {13}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{416 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(7 A-2 B) \sin \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(4 A+B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {3}{2} (e+f x)\right )}{32 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(22 A-7 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {5}{2} (e+f x)\right )}{160 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(A-4 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {7}{2} (e+f x)\right )}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {A (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {9}{2} (e+f x)\right )}{48 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(2 A-3 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {11}{2} (e+f x)\right )}{352 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {B (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {13}{2} (e+f x)\right )}{416 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

((7*A - 2*B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[
(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((4*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x]
)^2*(c - c*Sin[e + f*x])^(7/2))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x
)/2])^4) + ((22*A - 7*B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(160*f*(Cos[(
e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((A - 4*B)*Cos[(7*(e + f*x))/2]*(
a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x
)/2] + Sin[(e + f*x)/2])^4) + (A*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(48*f
*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2*A - 3*B)*Cos[(11*(e +
f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(352*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (B*Cos[(13*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(
7/2))/(416*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((7*A - 2*B)*S
in[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^
7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((4*A + B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[
(3*(e + f*x))/2])/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2
2*A - 7*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(5*(e + f*x))/2])/(160*f*(Cos[(e + f*x)/2] -
Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((A - 4*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e
+ f*x])^(7/2)*Sin[(7*(e + f*x))/2])/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^4) + (A*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(9*(e + f*x))/2])/(48*f*(Cos[(e + f*x
)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((2*A - 3*B)*(a + a*Sin[e + f*x])^2*(c -
c*Sin[e + f*x])^(7/2)*Sin[(11*(e + f*x))/2])/(352*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2]
+ Sin[(e + f*x)/2])^4) + (B*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(13*(e + f*x))/2])/(416*f*(
Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)

________________________________________________________________________________________

Maple [A]
time = 6.19, size = 121, normalized size = 0.58

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (\left (-1365 A +4935 B \right ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (11180 A -11820 B \right ) \sin \left (f x +e \right )+1155 B \left (\cos ^{4}\left (f x +e \right )\right )+\left (5915 A -10605 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )-12844 A +12204 B \right )}{15015 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/15015*(sin(f*x+e)-1)*c^4*(1+sin(f*x+e))^3*a^2*((-1365*A+4935*B)*sin(f*x+e)*cos(f*x+e)^2+(11180*A-11820*B)*si
n(f*x+e)+1155*B*cos(f*x+e)^4+(5915*A-10605*B)*cos(f*x+e)^2-12844*A+12204*B)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/
f

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^2*(-c*sin(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 375, normalized size = 1.79 \begin {gather*} \frac {2 \, {\left (1155 \, B a^{2} c^{3} \cos \left (f x + e\right )^{7} + 105 \, {\left (13 \, A - 14 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{6} + 35 \, {\left (91 \, A - 87 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} - 20 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 32 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} - 64 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right ) + 512 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} + {\left (1155 \, B a^{2} c^{3} \cos \left (f x + e\right )^{6} - 105 \, {\left (13 \, A - 25 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} + 140 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 160 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} + 192 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right ) + 512 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15015 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/15015*(1155*B*a^2*c^3*cos(f*x + e)^7 + 105*(13*A - 14*B)*a^2*c^3*cos(f*x + e)^6 + 35*(91*A - 87*B)*a^2*c^3*c
os(f*x + e)^5 - 20*(13*A - 3*B)*a^2*c^3*cos(f*x + e)^4 + 32*(13*A - 3*B)*a^2*c^3*cos(f*x + e)^3 - 64*(13*A - 3
*B)*a^2*c^3*cos(f*x + e)^2 + 256*(13*A - 3*B)*a^2*c^3*cos(f*x + e) + 512*(13*A - 3*B)*a^2*c^3 + (1155*B*a^2*c^
3*cos(f*x + e)^6 - 105*(13*A - 25*B)*a^2*c^3*cos(f*x + e)^5 + 140*(13*A - 3*B)*a^2*c^3*cos(f*x + e)^4 + 160*(1
3*A - 3*B)*a^2*c^3*cos(f*x + e)^3 + 192*(13*A - 3*B)*a^2*c^3*cos(f*x + e)^2 + 256*(13*A - 3*B)*a^2*c^3*cos(f*x
+ e) + 512*(13*A - 3*B)*a^2*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f
)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5985 deep

________________________________________________________________________________________

Giac [A]
time = 0.78, size = 392, normalized size = 1.87 \begin {gather*} -\frac {\sqrt {2} {\left (10010 \, A a^{2} c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 1155 \, B a^{2} c^{3} \cos \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, f x + \frac {13}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 60060 \, {\left (7 \, A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2 \, B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15015 \, {\left (4 \, A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) - 3003 \, {\left (22 \, A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 7 \, B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 4290 \, {\left (A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 4 \, B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) - 1365 \, {\left (2 \, A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 3 \, B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right )\right )} \sqrt {c}}{480480 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

-1/480480*sqrt(2)*(10010*A*a^2*c^3*cos(-9/4*pi + 9/2*f*x + 9/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 1155*B
*a^2*c^3*cos(-13/4*pi + 13/2*f*x + 13/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 60060*(7*A*a^2*c^3*sgn(sin(-1
/4*pi + 1/2*f*x + 1/2*e)) - 2*B*a^2*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1/2*f*x + 1/2*e) +
15015*(4*A*a^2*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + B*a^2*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-3
/4*pi + 3/2*f*x + 3/2*e) - 3003*(22*A*a^2*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 7*B*a^2*c^3*sgn(sin(-1/4*p
i + 1/2*f*x + 1/2*e)))*cos(-5/4*pi + 5/2*f*x + 5/2*e) + 4290*(A*a^2*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) -
4*B*a^2*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-7/4*pi + 7/2*f*x + 7/2*e) - 1365*(2*A*a^2*c^3*sgn(sin(-1
/4*pi + 1/2*f*x + 1/2*e)) - 3*B*a^2*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-11/4*pi + 11/2*f*x + 11/2*e)
)*sqrt(c)/f

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(7/2),x)

[Out]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(7/2), x)

________________________________________________________________________________________